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Standard Map

\begin{figure}\begin{center}\BoxedEPSF{standard_map050.epsf scaled 320}\BoxedEPSF{standard_map100.epsf scaled 320}\end{center}\end{figure}

\begin{figure}\begin{center}\BoxedEPSF{standard_map150.epsf scaled 320}\BoxedEPSF{standard_map200.epsf scaled 320}\end{center}\end{figure}

A 2-D Map, also called the Taylor-Greene-Chirikov Map in some of the older literature.

$\displaystyle I_{n+1}$ $\textstyle =$ $\displaystyle I_n+K\sin\theta_n$ (1)
$\displaystyle \theta_{n+1}$ $\textstyle =$ $\displaystyle \theta_n+I_{n+1} = I_n+\theta_n+K\sin\theta_n,$ (2)

where $I$ and $\theta$ are computed mod $2\pi$ and $K$ is a Positive constant. An analytic estimate of the width of the Chaotic zone (Chirikov 1979) finds
\delta I = B e^{-AK^{-1/2}}.
\end{displaymath} (3)

Numerical experiments give $A\approx 5.26$ and $B\approx 240$. The value of $K$ at which global Chaos occurs has been bounded by various authors. Greene's Method is the most accurate method so far devised.

Author Bound Fraction Decimal
Hermann $>$ ${1\over 34}$ 0.029411764
Italians $>$ - 0.65
Greene $\approx$ - 0.971635406
MacKay and Pearson $<$ ${63\over 64}$ 0.984375000
Mather $<$ ${4\over 3}$ 1.333333333

Fixed Points are found by requiring that

$\displaystyle I_{n+1}$ $\textstyle =$ $\displaystyle I_n$ (4)
$\displaystyle \theta_{n+1}$ $\textstyle =$ $\displaystyle \theta_n.$ (5)

The first gives $K\sin\theta_n=0$, so $\sin\theta_n=0$ and
\theta_n =0,\pi.
\end{displaymath} (6)

The second requirement gives
\end{displaymath} (7)

The Fixed Points are therefore $(I,\theta) =(0,0)$ and $(0, \pi)$. In order to perform a Linear Stability analysis, take differentials of the variables
$\displaystyle dI_{n+1}$ $\textstyle =$ $\displaystyle dI_n+K\cos\theta_n\,d\theta_n$ (8)
$\displaystyle d\theta_{n+1}$ $\textstyle =$ $\displaystyle dI_n+(1+K\cos\theta_n)\,d\theta_n.$ (9)

In Matrix form,
\delta I_{n+1}\cr \delta\theta_{n+1}\cr}}\r...
...ght]\left[{\matrix{\delta I_n\cr \delta \theta_n \cr}}\right].
\end{displaymath} (10)

The Eigenvalues are found by solving the Characteristic Equation
1-\lambda & K\cos\theta_n\cr
1 & 1+K\cos\theta_n-\lambda\cr}\right\vert=0,
\end{displaymath} (11)

\lambda^2-\lambda(K\cos\theta_n+2)+1 = 0
\end{displaymath} (12)

\lambda_\pm = {\textstyle{1\over 2}}[K\cos\theta_n+2\pm\sqrt{(K\cos\theta_n+2)^2-4}\,].
\end{displaymath} (13)

For the Fixed Point $(0, \pi)$,
$\displaystyle \lambda^{(0,\pi)}_\pm$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}[2-K\pm\sqrt{(2-K)^2-4}\,]$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(2-K\pm\sqrt{K^2-4K}\,).$ (14)

The Fixed Point will be stable if $\vert\Re(\lambda^{(0,\pi)})\vert<2.$ Here, that means
{\textstyle{1\over 2}}\vert 2-K\vert<1
\end{displaymath} (15)

\vert 2-K\vert<2
\end{displaymath} (16)

\end{displaymath} (17)

\end{displaymath} (18)

so $K\in [0,4)$. For the Fixed Point (0, 0), the Eigenvalues are
$\displaystyle \lambda^{(0,0)}_\pm$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}[2+K\pm\sqrt{(K+2)^2-4}\,]$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(2+K\pm\sqrt{K^2+4K}\,).$ (19)

If the map is unstable for the larger Eigenvalue, it is unstable. Therefore, examine $\strut\lambda^{(0,0)}_+$. We have
{1\over 2}\left\vert{2+K+\sqrt{K^2+4K}\,}\right\vert<1,
\end{displaymath} (20)

\end{displaymath} (21)

\end{displaymath} (22)

But $K>0$, so the second part of the inequality cannot be true. Therefore, the map is unstable at the Fixed Point (0, 0).


Chirikov, B. V. ``A Universal Instability of Many-Dimensional Oscillator Systems.'' Phys. Rep. 52, 264-379, 1979.

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© 1996-9 Eric W. Weisstein